Excellent Abstract Elementary Classes are tame

The assumption that an AEC is tame is a powerful assumption permitting development of stability theory for AECs with the amalgamation property. Lately several upward categoricity theorems were discovered where tameness replaces strong set-theoretic assumptions. We present in this article two sufficient conditions for tameness, both in form of strong amalgamation properties that occur in nature. One of them was used recently to prove that several Hrushovski classes are tame. This is done by introducing the property of weak $(\mu,n)$-uniqueness which makes sense for all AECs (unlike Shelah's original property) and derive it from the assumption that weak (\LS(\K),n)-uniqueness, (\LS(\K),n)-symmetry and (\LS(\K),n)-existence properties hold for all $n<\omega$. The conjunction of these three properties we call \emph{excellence}, unlike \cite{Sh 87b} we do not require the very strong (\LS(\K),n)-uniqueness, nor we assume that the members of \K are atomic models of a countable first order theory. We also work in a more general context than Shelah's good frames.Comment: 26 page

Short-time critical dynamics of the three-dimensional systems with long-range correlated disorder

Monte Carlo simulations of the short-time dynamic behavior are reported for three-dimensional Ising and XY models with long-range correlated disorder at criticality, in the case corresponding to linear defects. The static and dynamic critical exponents are determined for systems starting separately from ordered and disordered initial states. The obtained values of the exponents are in a good agreement with results of the field-theoretic description of the critical behavior of these models in the two-loop approximation and with our results of Monte Carlo simulations of three-dimensional Ising model in equilibrium state.Comment: 24 RevTeX pages, 12 figure